\documentclass[aps,twocolumn]{revtex4}
\usepackage{graphicx}
\usepackage{amssymb,amsfonts,amsmath,amsthm}
\usepackage{chemarr}
\usepackage{bm}
\usepackage{pslatex}
\usepackage{mathptmx}
\usepackage{xfrac}

%% concentration notations
\newcommand{\mymat}[1]{\boldsymbol{#1}}
\newcommand{\mytrn}[1]{{#1}^{\mathsf{T}}}
\newcommand{\myvec}[1]{\overrightarrow{#1}}
\newcommand{\mygrad}{\vec{\nabla}}
\newcommand{\myhess}{\mathcal{H}}
\newcommand{\myd}{\mathrm{d}}

\begin{document}

\section{Continuous function approximation}
Let $g(x)$ be a function that we want to approximate by $f\left(x,\vec{a}\right)$ on an interval $I$,
and $\vec{a}\in \mathbb{R}^N$.

We define
\begin{equation}
	D^2\left(\vec{a}\right) =
	 \dfrac{1}{2} \int_I \left\lbrack g(x) - f\left(x,\vec{a}\right) \right\rbrack^2 \; \myd x.
\end{equation}
We obtain a set of $N$ equations
\begin{equation}
	 \dfrac{\partial D^2}{ \partial a_i } = - \int_I \left\lbrack g(x) - f\left(x,\vec{a}\right) \right\rbrack \dfrac{\partial f}{\partial a_i} \; \myd x.
\end{equation}
We want to solve, for all $i$
\begin{equation}
	G_i\left(\vec{a}\right) =\int_I \left\lbrack g(x) - f\left(x,\vec{a}\right) \right\rbrack \dfrac{\partial f}{\partial a_i} \; \myd x.
\end{equation}
Starting from $\vec{G}_n\left(\vec{a}\right)$, we approximate the opposite Jacobian of $\vec{G}$ with
\begin{equation}
	\mymat{J} = \int_I \left(\dfrac{\partial f}{\partial a_i}\right)\left(\dfrac{\partial f}{\partial a_j}\right)\; \myd x.
\end{equation}

\end{document}